Napierian logarithm

${\displaystyle \mathrm{N}\mathrm{a}\mathrm{p}\mathrm{L}\mathrm{o}\mathrm{g}(x)=-{10}^7\ln (x/{10}^7)}$

The Napierian logarithm satisfies identities quite similar to the modern logarithm, such as

${\displaystyle \mathrm{N}\mathrm{a}\mathrm{p}\mathrm{L}\mathrm{o}\mathrm{g}(xy)\approx \mathrm{N}\mathrm{a}\mathrm{p}\mathrm{L}\mathrm{o}\mathrm{g}(x)+\mathrm{N}\mathrm{a}\mathrm{p}\mathrm{L}\mathrm{o}\mathrm{g}(y)-161180956}$

or

${\displaystyle \mathrm{N}\mathrm{a}\mathrm{p}\mathrm{L}\mathrm{o}\mathrm{g}(xy/{10}^7)=\mathrm{N}\mathrm{a}\mathrm{p}\mathrm{L}\mathrm{o}\mathrm{g}(x)+\mathrm{N}\mathrm{a}\mathrm{p}\mathrm{L}\mathrm{o}\mathrm{g}(y)}$

In Napier's 1614 Mirifici Logarithmorum Canonis Descriptio, he provides tables of logarithms of sines for 0 to 90°, where the values given (columns 3 and 5) are

${\displaystyle \mathrm{N}\mathrm{a}\mathrm{p}\mathrm{L}\mathrm{o}\mathrm{g}(\theta )=-{10}^7\ln (\sin (\theta ))}$

Properties

Napier's "logarithm" is related to the natural logarithm by the relation

${\displaystyle \mathrm{N}\mathrm{a}\mathrm{p}\mathrm{L}\mathrm{o}\mathrm{g}(x)\approx 10000000(16.11809565-\ln x)}$

and to the common logarithm by

${\displaystyle \mathrm{N}\mathrm{a}\mathrm{p}\mathrm{L}\mathrm{o}\mathrm{g}(x)\approx 23025851(7-{\log }_{10}x).}$

Note that

${\displaystyle 16.11809565\approx 7\ln \left(10\right)}$

and

${\displaystyle 23025851\approx {10}^7\ln (10).}$

Napierian logarithms are essentially natural logarithms with decimal points shifted 7 places rightward and with sign reversed. For instance the logarithmic values

${\displaystyle \ln (.5000000)=-0.6931471806}$

${\displaystyle \ln (.3333333)=-1.0986123887}$

would have the corresponding Napierian logarithms:

${\displaystyle \mathrm{N}\mathrm{a}\mathrm{p}\mathrm{L}\mathrm{o}\mathrm{g}(5000000)=6931472}$

${\displaystyle \mathrm{N}\mathrm{a}\mathrm{p}\mathrm{L}\mathrm{o}\mathrm{g}(3333333)=10986124}$

For further detail, see history of logarithms.